Conference Agenda

The classification of movement in space is one of the key tasks in environmental science. Various geospatial data such as rainfall or other weather data, data on animal movement or landslide data require a quantitative analysis of the probable movement in space to obtain information on potential risks, ecological developments or changes in future. Usually, machine-learning tools are applied for this task. Yet, machine-learning approaches also have some drawbacks, e.g. the often required large training sets and the fact that the algorithms are often hard to interpret. We propose a classification approach for spatial data based on ordinal patterns. Ordinal patterns have the advantage that they are easily applicable, even to small data sets, are robust in the presence of certain changes in the time series and deliver interpretative results. They, therefore, do not only offer an alternative to machine-learning in the case of small data sets but might also be used in pre-processing for a meaningful feature selection. In this talk, we introduce the basic concept of multivariate ordinal patterns, classify them and provide the corresponding limit theorem. We focus on the discrete case, that is, on movements on a two dimensional grid. The approach is applied to rainfall radar data.

Since their introduction by Bandt and Pompe (2002), ordinal patterns have become a popular tool for dynamical systems, statistics and data analysis. As the name may already suggest, ordinal patterns capture the ordinal structure of the underlying data within a moving window. They have many desirable properties like invariance under monotone transformations, robustness with respect to small noise and simplicity in application. In particular, ordinal patterns are able to capture possibly non-linear dependence.

These properties are directly transferred to ordinal pattern based measures, e.g., permutation entropy. Since permutation entropy is defined as the Shannon entropy of the ordinal pattern distribution, it is a natural idea to also consider other variants of complexity measures based on ordinal patterns. Here, we particularly consider a variant based on Renyi-2 entropy, which is strongly related to the symbolic correlation integral recently proposed by Caballero-Pintado et al. (2019).

We derive the limit distribution of the symbolic correlation integral (and hence also the Renyi-2 entropy) for a broad class of short-range dependent processes, namely 1-approximating functionals. Therefore, we complement the results by Caballero-Pintado et al. (2019) who only considered the i.i.d.-case. Our contributions prove to be useful in a variety of classification tasks, that is, they allow us to distinguish time series or data stemming from time series based on the degree of complexity present in each one of them. For example, to a certain extent we are able to distinguish, e.g., ARMA-models that differ only in the choice of their parameters. Furthermore, our approach allows for testing whether two time series follow the same underlying model in the sense of a distinction between, e.g., an AR- and an MA-model. We present a selection of our simulation study and illustrate the applicability of our test with a real-world data example given by the Bonn EEG-data base.

For real-valued time series, the autoregressive moving-average (ARMA) models are of utmost relevance in practice, not only because of their own modeling abilities, but also because they constitute the core of innumerable further time series models. While ARMA models are not directly applicable to discrete-valued time series, they served as an inspiration for defining several ``ARMA-like'' models for such data. An example is the so-called NDARMA model (new discrete ARMA), which constitutes a universally applicable ARMA-like model for any type of time series data. However, its generated sample paths are characterized by the repeated occurrence of one and the same value (``runs'') being interrupted by sudden jumps. Therefore, the NDARMA model is mainly relevant for nominal time series in practice, as the aforementioned features of the sample paths are hardly observed for real-world ordinal or even quantitative time series.

In this talk, a new and flexible class of ARMA-like models for nominal or ordinal time series is presented, which are characterized by using so-called ``weighting operators'' and are, thus, referred to as weighted discrete ARMA (WDARMA) models. By choosing an appropriate type of weighting operator, one can model, for example, nominal time series with negative serial dependencies, or ordinal time series where transitions to neighbouring states are more likely than sudden large jumps. Essential stochastic properties of WDARMA models are derived, such as the existence of a stationary, ergodic, and $\varphi$-mixing solution as well as closed-form formulae for marginal and bivariate probabilities. Numerical illustrations as well as simulation experiments regarding the finite-sample performance of maximum likelihood estimation are presented. The possible benefits of using an appropriate weighting scheme within the WDARMA class are demonstrated by an application to an ordinal time series on the air quality in Beijing.

Reference:

Weiß, C.H., Swidan, O. (2024) Weighted discrete ARMA models for categorical time series. \textit{Journal of Time Series Analysis}, in press. DOI: https://doi.org/10.1111/jtsa.12773

For discrete-valued time series, predictive inference can not be implemented through the construction of prediction intervals to some pre-determined coverage level, as this is the case for real-valued time series. Although prediction sets rather than intervals respect more the discrete nature of the data and appear to be more natural, they are generally not able to retain a desired coverage level neither in finite samples nor asymptotically. To address this general problem of predictive inference for discrete-valued time series, we propose to reverse the construction principle by considering pre-defined sets of interest and estimating the corresponding predictive probability, that is, the probability that a future observation falls in these sets given the time series observed. The accuracy of the corresponding prediction procedure is then evaluated by quantifying the uncertainty associated with estimation of this predictive probability. For this purpose, parametric and non-parametric approaches are considered and asymptotic theory for the estimators involved is derived. Since the established limiting distributions are typically cumbersome to apply in practice, we propose suitable bootstrap approaches to evaluate the distribution of the estimators used. These bootstrap approaches also have the advantage to imitate the distributions of interest under different possible settings including the important case where a misspecified model is used for prediction. Considering such different settings leads to confidence intervals for the predictive probability which properly take into all sources of uncertainty that affect prediction. We elaborate on bootstrap implementations under different scenarios and we focus on the case of INAR and INARCH models. Simulations investigate the finite sample performance of the methods developed by considering different parametric and non-parametric bootstrap implementations to account for the various sources of randomness and variability. Applications to real life data sets are also presented.